Good Writing on Exponents

There's been a lot of good writing about exponents recently, some of it in response to my most recent post. Here's a sampling:

• Christopher Danielson wonders whether the problem with exponents is that we introduce it as repeated multiplication. He's thinking that "number of doublings" might be better, and he tests this theory on his kid. (The comments on the post are great also. Check them out.)
• Chris Robinson modified the exponents survey for his own students, collected a ton of responses and offered his own analysis of what kids are doing with exponents. I still have to dig into his students' survey responses carefully, but I think that my favorite snippet so far is this:

         Exponentiation sometimes defaults to multiplication, and multiplication sometimes defaults to addition.

• ... and some more survey results! Thanks Mrs Reilly! This effect is for realz, guys.
• Andrew Stadel wraps it up with some extremely solid lesson plans that put these mistakes right in front of the kids noses. His theory? Part of the way to change kids intuitions about this stuff is to draw out and directly challenge their previous ideas. I stole his stuff for class, and it went quite well. 

Exponents are cool.